1. Lesson 1 - 1 Displaying Distribution with Graphs

2. Histograms Histograms break the range of data values into classes and displays the count or % of observations that fall into that class –Divide the range of data into equal-width classes –Count the observations in each class: “frequency” –Draw bars to represent classes: height = frequency –Bars should touch (unlike bar graphs).

3. Histogram versus Bar Chart HistogramBar Chart variablesquantitativecategorical bar spaceno spacespaces between

4. Determining Classes and Widths The number of classes k to be constructed can be roughly approximated by k =  number of observations To determine the width of a class use max - min w = ----------------- k and always round up to the same decimal units as the original data.

5. Example 1 The ages (measured by last birthday) of the employees of Dewey, Cheatum and Howe are listed below. a)Construct a stem graph of the ages b)Construct a back-to-back comparing the offices c)Construct a histogram of the ages 223121492642 30283139 203732363533 454749382848 Office A Office B

6. Example 1 cont n = 24 k = √24 ≈ 4.9 so pick k = 5 w = (49 – 20)/5 = 29/5 ≈ 5.8  6 K rangeNr 120 – 253 226 – 316 332 – 375 438 – 435 544 – 50 5 2 4 6 8 20-25 26-31 32-37 38-43 44-50 Numbers of Personnel Ages

7. Example 1 cont n = 24 k = √24 ≈ 4.9 so pick k = 5 w = (49 – 20)/5 = 29/5 ≈ 5.8  6 K rangeNr 120 – 253 226 – 316 332 – 375 438 – 435 544 – 50 5 2 4 6 8 20 26 32 38 44 50 Numbers of Personnel Ages

8. Example 1: Histogram n = 24 k = √24 ≈ 4.9 so pick k = 4 w = (49 – 20)/4 = 29/4 ≈ 7.3  8 K rangeNr 120 – 274 228 – 358 336 – 437 444 – 515 2 4 6 8 20-27 27-35 36-43 44-51 Numbers of Personnel Ages

9. Example 2 Below are times obtained from a mail-order company's shipping records concerning time from receipt of order to delivery (in days) for items from their catalogue? a)Construct a stem plot of the delivery times b)Construct a split stem plot of the delivery times c)Construct a histogram of the delivery times 371051412 629222511 5712102223 14854713 2731132168 3101912118

10. Example 2: Histogram n = 36 k = √36 = 6 w = (31 – 2)/6 = 29/6 ≈ 4.8  5 K range1Nr 12 – 69 27 – 1112 312 – 167 417 – 212 522 – 26 4 627 – 312 2 4 6 8 2 7 12 17 22 27 32 Frequency Days to Delivery 10 12

11. Exploratory Data Analysis Exploratory Data Analysis (EDA): –Statistical practice of analyzing distributions of data through graphical displays and numerical summaries. Distribution: –Description of the values a variable takes on and how often the variable takes on those values. An EDA allows us to identify patterns and departures from patterns in distributions.

12. Describing Distributions Overall patterns of a distribution should be described by anything unusual and: –Shape of its graph symmetric, skewed, unimodal, bimodal, etc –Center Quantitative: mean (symmetric data) median (skewed data) Categorical: mode –Spread Quantitative: range, standard deviation, IQR

13. Uniform Mound-like (Bell-Shaped) Skewed Left (-- tail) Bi-Modal Skewed Right (-- tail) Frequency Distributions

14. Exploratory Data Analysis Summary The purpose of an EDAis to organize data and identify patterns/departures. PLOT YOUR DATA –Choose an appropriate graph Look for overall pattern and departures from pattern –Shape {mound, bimodal, skewed, uniform} –Outliers {points clearly away from body of data} –Center {What number “typifies” the data?} –Spread {How “variable” are the data values?}

15. Time Series Plot Time on the x-axis Interested values on the y-axis Look for seasonal (periodic) trends in data –What seasonal trends do you expect in the following chart?

16. Ave Gas Prices Time Series Plot

17. Seasonal Trends Gas prices go up during the summer –Memorial Day to Labor Day Sharp increases with Hurricane activity –Hurricane season generally July – October Major supply issues cause sharp increases Positive general increase (due to inflation)

18. Cautions Label all axeses and title all graphs Histogram rectangles touch each other; rectangles in bar graphs do not touch. Can’t have class widths that overlap Raw data can be retrieved from the stem-and-leaf plot; but a frequency distribution of histogram of continuous data summarizes the raw data Only quantitative data can be described as skewed left, skewed right or symmetric (uniform or bell- shaped)

19. Day 2 Summary and Homework Summary –Examining a Distribution: Shape, Outliers, Center, Spread Shape: Symmetric, Skewed, xx-modal Outliers: Judgment (Rule coming) Center: Mean and Median Spread: Standard Deviation, IQR, and Range –Histograms, stemplots and dot plots allow distribution examination –Time plots (look at seasonal trends) Homework –pg 55-58 probs 8-12 and pg 65 prob 16